Entropy stable finite difference schemes for Chew, Goldberger & Low anisotropic plasma flow equations

TL;DR

This work develops high-order entropy-stable finite-difference schemes for the CGL anisotropic plasma flow equations by reformulating the system to decouple non-conservative terms from entropy, applying Godunov symmetrization to the conservative part, and building entropy-conservative fluxes with judicious diffusion based on entropy-scaled eigenvectors. The approach yields semi-discrete entropy stability and is extended to higher-order accuracy via sign-preserving reconstruction, with explicit SSP-RK time stepping for anisotropic tests and ARK-IMEX schemes for isotropic limits. Numerical results across 1D and 2D test problems—including Brio–Wu, Ryu–Jones, multiple Riemann problems, and Orszag–Tang vortex—show accurate wave resolution, proper entropy decay, and agreement with MHD in the isotropic limit. The framework advances robust, provably stable simulations of non-conservative hyperbolic plasmas and provides a foundation for future physics-informed enhancements in kinetic regimes.

Abstract

In this article, we consider the Chew, Goldberger \& Low (CGL) plasma flow equations, which is a set of nonlinear, non-conservative hyperbolic PDEs modelling anisotropic plasma flows. These equations incorporate the double adiabatic approximation for the evolution of the pressure, making them very valuable for plasma physics, space physics and astrophysical applications. We first present the entropy analysis for the weak solutions. We then propose entropy-stable finite-difference schemes for the CGL equations. The key idea is to rewrite the CGL equations such that the non-conservative terms do not contribute to the entropy equations. The conservative part of the rewritten equations is very similar to the magnetohydrodynamics (MHD) equations. We then symmetrize the conservative part by following Godunov's symmetrization process for MHD. The resulting equations are then discretized by designing entropy conservative numerical flux and entropy diffusion operator based on the entropy scaled eigenvectors of the conservative part. We then prove the semi-discrete entropy stability of the schemes for CGL equations. The schemes are then tested using several test problems derived from the corresponding MHD test cases.

Figures (9)

• Figure 1: \ref{['test:bw']}: Plots of density, parallel and perpendicular pressure components, and total entropy decay at each time step for explicit schemes without source term and IMEX scheme with source term using $2000$ cells at final time $t = 0.2$.
• Figure 2: \ref{['test:ryu']}: Plots of density, parallel and perpendicular pressure components, and total entropy decay at each time step for explicit schemes without source term and IMEX scheme with source term using $2000$ cells at final time $t = 0.2$.
• Figure 3: \ref{['test:sf']}: Plots of density, parallel and perpendicular pressure components, and total entropy decay at each time step for explicit schemes without source term and IMEX scheme with source term using $2000$ cells at final time $t = 0.05$.
• Figure 4: \ref{['test:rp4']}: Plots of density, parallel and perpendicular pressure components, and total entropy decay at each time step for explicit schemes without source term and IMEX scheme with source term using $2000$ cells at final time $t = 0.15$.
• Figure 5: \ref{['test:rp5']}: Plots of density, parallel and perpendicular pressure components, and total entropy decay at each time step for explicit schemes without source term and IMEX scheme with source term using $2000$ cells at final time $t = 0.15$.

Theorems & Definitions (16)

• lemma thmcounterlemma
• proof
• remark thmcounterremark
• definition thmcounterdefinition
• remark thmcounterremark
• definition thmcounterdefinition
• Theorem 3.1
• Theorem 4.1
• proof
• Theorem 4.2